# Ratio of sums vs sum of ratio

Is anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating

$A(x,y)= \frac{\sum_z f(x,y,z)}{\sum_z g(y,z)}$

and

$B(x,y)= \sum_z\left(\frac{f(x,y,z)}{g(y,z)}\right)$

?

Specific context (for what I'm dealing with, but not necessarily the question) is that $\sum_{x,y,x}f(x,y,z)=1$ and $\sum_{y,x}g(y,z)=1$ and $f(x,y,z)\geq 0$ and $g(y,z)\geq 0\quad \forall x,y,z$. I.e. probabilities (or more generally, I guess, measures).

It seems like it could 'vaguely' be related to log sum inequalities (when transformed) or Jensen's inequality perhaps?

If you assume that both $$f$$ and $$g$$ are nonnegative, you have $$A(x, y) \leq B(x, y)$$.

Proof:

$$\frac{f(x, y, z)}{\sum_{z}g(y, z)} \leq \frac{f(x,y,z)}{g(y,z)}$$, since $$g(y, z) \leq \sum_{z}g(y, z)$$.

So $$A(x,y) = \frac{\sum_z f(x,y,z)}{\sum_z g(y,z)} = \sum_z \frac{f(x, y, z)}{\sum_z g(y, z)} \leq \sum_z \frac{f(x, y, z)}{g(y, z)} = B(x,y)$$.

One example would be Jensen's inequality:

https://en.wikipedia.org/wiki/Jensen%27s_inequality

• I'm not entirely clear how that applies directly, can you elaborate? Broadly, Jensen's equality seems to switch the order of the summation and application of convex function, not between different summations... Jan 21 '16 at 4:32